388 Cosmic microwave background anisotropies
changes in the baryon and cold dark matter densities. If the baryon density is made
too large, though, an increase in (^) mh^275 cannot compensate an increase in (^) bh^275
because its effect onH 1 saturates for large values (and moreover (^) mh^275 cannot be
much greater than unity).
Current observations suggest thatH 1 is 6–8 times the amplitude at large angular
scales.Based on the peak height alone, we can be sure that the baryon density is less
than20% of the critical density. Although much better constraints can be obtained
using the full power spectrum and additional data, it is important to appreciate that
the peak height alone suffices to rule out a flat, baryon-dominated universe, which
was, in essence, the original concept of the big bang universe.
If the height of the first peak is kept fixed, the only freedom left is a simultaneous
change of the cold matter and baryon densities, both of which can be either increased
or decreased. For instance, we can still keep the height of the first peak unchanged
simultaneouslyincreasing the baryon and matter density by a factor of about 1.5
from the concordance model densities. However, since the increases of the baryon
and cold matter densities have opposing effects on the location of the peak, its net
shift will be negligible. This explains why, for a fixed height, the location of the
first peak depends sensitively on the spatial curvature only, allowing us to resolve
the degeneracy in its determination. The current data on the location of the first
peak strongly support the flat universe predicted by inflation.
To break the degeneracy between baryon and matter density altogether, it suffices
to consider the second acoustic peak, which results primarily from the destructive
interference between oscillatory terms in (9.102), but bearing in mind that they are
superimposed on the “hill” due toN. The first term in (9.102) makes a negative
contribution to the second peak and has a coefficient that is proportional toξ.
The second term makes a positive contribution to the second peak but slightly
decreases asξincreases. Hence, one can see that the second peak shrinks as the
baryon density increases. The two terms nearly cancel altogether when the baryon
density is about 8% of the critical density, or about twice the best-fit value. Curiously
enough, though, exact numerical calculations show that a tiny second peak survives
for (^) mh^275 0 .26 even when the baryon density is made much greater than 8%.
This is because an increase in the baryon density also increasesN 1 , making the
hill on which the oscillatory contributions rest much steeper in the vicinity of the
second peak. In other words, the appearance of the second peak depends on a
delicate cancellation and combination of diverse terms. So, for example, it would
be incorrect to conclude that the baryon density is less than 8% of the critical density
simply because one observes a second acoustic peak.
However, combining information about the height of the first peak with the fact
that the second peakexists(ignoring peak locations for the moment) does lead
to good limits on both baryon density and cold dark matter density. We initially